Problem Set #2
A. W. Stetz
October 11, 2007
These problems are due Friday, Oct. 19.
1. The generating function expansion for the Legendre polynomials is
V= (1 −2hz +h2)−1/2=
∞
X
l=0
hlPl(z)
where 0 <h<1 and z= cos(θ). By expanding the left side of this
equation in powers of h, prove that
Pl(z) =
rmax
X
r=0
(−1)r(2l−2r)!
2lr!(l−r)!(l−2r)!zl−2r
where rmax =l/2 or (l−1)/2, whichever is an integer.
2. Prove that the polynomials Pl(z) given by the formula above satisfy
Legendre’s equation
d
dz (1 −z2)dPl
dz +l(l+ 1)Pl= 0
3. Given the generating function above
(a) Verify the formula
(1 −2hz +h2)∂V
∂h = (z−h)V
and show by comparing powers of hon both sides of the equation
that
(l+ 1)Pl+1 −(2l+ 1)zPl+lPl−1= 0
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